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1、Chapter ThirteenRisky AssetsMean of a DistributionuA random variable (r.v.) w takes values w1,wS with probabilities 1,., S ( 1 + + S = 1).uThe mean (expected value) of the distribution is the av. value of the r.v.;E E .wwws ssS1 1Variance of a DistributionuThe distributions variance is the r.v.s av.
2、 squared deviation from the mean;uVariance measures the r.v.s variation. varvar ().wwwswssS2 22 21 1Standard Deviation of a DistributionuThe distributions standard deviation is the square root of its variance;uSt. deviation also measures the r.v.s variability. st.st. dev dev ().wwwwswssS2 22 21 1Mea
3、n and VarianceProbabilityRandom Variable ValuesTwo distributions with the samevariance and different means.Mean and VarianceProbabilityRandom Variable ValuesTwo distributions with the samemean and different variances.Preferences over Risky AssetsuHigher mean return is preferred.uLess variation in re
4、turn is preferred (less risk).Preferences over Risky AssetsuHigher mean return is preferred.uLess variation in return is preferred (less risk).uPreferences are represented by a utility function U( , ).uU as mean return .uU as risk .Preferences over Risky AssetsPreferredHigher mean return is a good.H
5、igher risk is a bad.Mean Return, St. Dev. of Return, Preferences over Risky AssetsPreferredHigher mean return is a good.Higher risk is a bad.Mean Return, St. Dev. of Return, Preferences over Risky AssetsuHow is the MRS computed?Preferences over Risky AssetsuHow is the MRS computed?dUUdUdUdUdddUU 0 0
6、/.Preferences over Risky AssetsMean Return, St. Dev. of Return, PreferredHigher mean return is a good.Higher risk is a bad.ddUU /Budget Constraints for Risky AssetsuTwo assets.uRisk-free assets rate-or-return is rf .uRisky stocks rate-or-return is ms if state s occurs, with prob. s .uRisky stocks me
7、an rate-of-return isrmms ssS.1 1Budget Constraints for Risky AssetsuA bundle containing some of the risky stock and some of the risk-free asset is a portfolio.ux is the fraction of wealth used to buy the risky stock.uGiven x, the portfolios av. rate-of-return isrxrx rxmf().1 1Budget Constraints for
8、Risky Assetsrxrx rxmf().1 1x = 0 rrxfand x = 1 rrxm.Budget Constraints for Risky Assetsrxrx rxmf().1 1x = 0 rrxfand x = 1 rrxm.Since stock is risky and risk is a bad, for stockto be purchased must haverrmf.Budget Constraints for Risky Assetsrxrx rxmf().1 1x = 0 rrxfand x = 1 rrxm.Since stock is risk
9、y and risk is a bad, for stockto be purchased must haverrmf.So portfolios expected rate-of-return rises with x(more stock in the portfolio).Budget Constraints for Risky AssetsuPortfolios rate-of-return variance isxsfxssSxmx rr2 22 21 11 1().Budget Constraints for Risky AssetsuPortfolios rate-of-retu
10、rn variance isxsfxssSxmx rr2 22 21 11 1().rxrx rxmf().1 1Budget Constraints for Risky AssetsuPortfolios rate-of-return variance isxsfxssSxmx rr2 22 21 11 1().rxrx rxmf().1 1xsfmfssSxmx rxrx r2 22 21 11 11 1()()Budget Constraints for Risky AssetsuPortfolios rate-of-return variance isxsfxssSxmx rr2 22
11、 21 11 1().rxrx rxmf().1 1xsfmfssSsmssSxmx rxrx rxmxr2 22 21 12 21 11 11 1()()()Budget Constraints for Risky AssetsuPortfolios rate-of-return variance isxsfxssSxmx rr2 22 21 11 1().rxrx rxmf().1 1xsfmfssSsmssSsmssSxmx rxrx rxmxrxmr2 22 21 12 21 12 22 21 11 11 1()()()()Budget Constraints for Risky As
12、setsuPortfolios rate-of-return variance isxsfxssSxmx rr2 22 21 11 1().rxrx rxmf().1 1xsfmfssSsmssSsmssSmxmx rxrx rxmxrxmrx2 22 21 12 21 12 22 21 12 22 21 11 1()()()().Budget Constraints for Risky Assetsxmx2 22 22 2Variancexmx.so st. deviationBudget Constraints for Risky Assetsxmx2 22 22 2x = 0 and x
13、 = 1 x 0 0 xm.Variancexmx.so st. deviationBudget Constraints for Risky Assetsxmx2 22 22 2x = 0 and x = 1 x 0 0 xm.Variancexmx.so st. deviationSo risk rises with x (more stock in the portfolio). Budget Constraints for Risky AssetsMean Return, St. Dev. of Return, Budget Constraints for Risky Assetsrxr
14、x rxmf().1 1xmx.xrrxfx0 00 0,0 0rfMean Return, St. Dev. of Return, Budget Constraints for Risky Assetsrxrx rxmf().1 1xmx.xrrxfx0 00 0,m0 0rmxrrxmxm1 1,rfMean Return, St. Dev. of Return, Budget Constraints for Risky Assetsrxrx rxmf().1 1xmx.xrrxfx0 00 0,m0 0rmxrrxmxm1 1,Budget linerfMean Return, St.
15、Dev. of Return, Budget Constraints for Risky Assetsrxrx rxmf().1 1xmx.xrrxfx0 00 0,m0 0rmxrrxmxm1 1,Budget line, slope = rfrrmfmMean Return, St. Dev. of Return, Choosing a Portfoliom0 0rmBudget line, slope = rfrrmfmMean Return, St. Dev. of Return, is the price of risk relative tomean return.Choosing
16、 a Portfoliom0 0rmBudget line, slope = rfrrmfmWhere is the most preferredreturn/risk combination?Mean Return, St. Dev. of Return, Choosing a Portfoliom0 0rmBudget line, slope = rfrrmfmWhere is the most preferredreturn/risk combination?Mean Return, St. Dev. of Return, Choosing a Portfoliom0 0rmBudget
17、 line, slope = rfrrmfmWhere is the most preferredreturn/risk combination?rxxMean Return, St. Dev. of Return, Choosing a Portfoliom0 0rmBudget line, slope = rfrrM RSmfmWhere is the most preferredreturn/risk combination?rxxMean Return, St. Dev. of Return, Choosing a Portfoliom0 0rmBudget line, slope =
18、 rfrrUUmfm /Where is the most preferredreturn/risk combination?rxxMean Return, St. Dev. of Return, Choosing a PortfoliouSuppose a new risky asset appears, with a mean rate-of-return ry rm and a st. dev. y m.u Which asset is preferred?Choosing a PortfoliouSuppose a new risky asset appears, with a mea
19、n rate-of-return ry rm and a st. dev. y m.u Which asset is preferred?uSupposerrrryfymfm.Choosing a Portfoliom0 0rmrfrxxBudget line, slope = rrmfmMean Return, St. Dev. of Return, Choosing a Portfoliom0 0rmBudget line, slope = rfrrmfmrxxryyMean Return, St. Dev. of Return, Choosing a Portfoliom0 0rmrfB
20、udget line, slope = rrmfmrxxryyBudget line, slope = rryfyMean Return, St. Dev. of Return, Choosing a Portfoliom0 0rmrfBudget line, slope = rrmfmrxxryyBudget line, slope = rryfyHigher mean rate-of-return andhigher risk chosen in this case.Mean Return, St. Dev. of Return, Measuring RiskuQuantitatively
21、, how risky is an asset?uDepends upon how the assets value depends upon other assets values.uE.g. Asset As value is $60 with chance 1/4 and $20 with chance 3/4.uPay at most $30 for asset A.Measuring RiskuAsset As value is $60 with chance 1/4 and $20 with chance 3/4.uAsset Bs value is $20 when asset
22、As value is $60 and is $60 when asset As value is $20 (perfect negative correlation of values).uPay up to $40 $30 for a 50-50 mix of assets A and B.Measuring RiskuAsset As risk relative to risk in the whole stock market is measured byA Ari sk of asri sk of asset Aset Ari sk of w hri sk of w hol e m
23、arketol e m arket.Measuring RiskuAsset As risk relative to risk in the whole stock market is measured byA Ari sk of asri sk of asset Aset Ari sk of w hri sk of w hol e m arketol e m arket.A AA Acovari ancecovari ance( (vari ance(vari ance(rrrmm,)where is the markets rate-of-returnand is asset As rat
24、e-of-return.rA ArmMeasuring Risku asset As return is not perfectly correlated with the whole markets return and so it can be used to build a lower risk portfolio. 1 11 1A A.A A 1 1Equilibrium in Risky Asset MarketsuAt equilibrium, all assets risk-adjusted rates-of-return must be equal.uHow do we adj
25、ust for riskiness?Equilibrium in Risky Asset MarketsuRiskiness of asset A relative to total market risk is A. uTotal market risk is m.uSo total riskiness of asset A is A m.Equilibrium in Risky Asset MarketsuRiskiness of asset A relative to total market risk is A. uTotal market risk is m.uSo total ri
26、skiness of asset A is A m.uPrice of risk isuSo cost of asset As risk is p A m.prrmfm.Equilibrium in Risky Asset MarketsuRisk adjustment for asset A isuRisk adjusted rate-of-return for asset A isprrrrmmfmmmf A AA AA A().rrrmfA AA A().Equilibrium in Risky Asset MarketsuAt equilibrium, all risk adjuste
27、d rates-of-return for all assets are equal.uThe risk-free assets = 0 so its adjusted rate-of-return is just uHence, for every risky asset A.rrrrrrrrfmffmfA AA AA AA Ai . i .e.e. ()()rf.Equilibrium in Risky Asset MarketsuThat at equilibrium in asset markets is the main result of the Capital Asset Pricing Model (CAPM), a model used extensively to study financial markets.rrrrfmfA AA A()